The word "adapter" or adaptor is (IMHO) a sort of generic or proprietary device term, that does however imply a conversion of a factory or cut pipe spigot end to an in some configuration simulated flanged end (e.g. to allow a flanged piping or valve etc. item to be inserted into the line, and/or perhaps also in "field-adaptable" or cut-to-suit fashion, to also precisely locate the flanged end at a particular station on the pipeline).
While there are actually traditional, standardized fittings per first ANSI/AWWA C110 and then C153 standards called mechanical joint by flanged "connecting pieces", that can be used to do basically the same thing, some manufacturers/fabricators have developed various more compact and proprietary design "adapters" to basically serve the same function, but with much lesser (e.g. casting) weight and laying length involved in the transition at the plain end(s). These devices generally employ a sealing mechanism e.g. actuated by the eventual flanged bolting, and as traditional flanged ends are also restrained joints (and sometimes used e.g. at substantial thrust foci such as block or isolation valves etc.), also often some means of mechanically attaching the device to the plain end to be able to handle some level of axial thrust. The specifier should consider the designs, strength and also where necessary deflection abilities while maintaining seal (etc. compared to traditional flanges) of all such devices.